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Quantum random walks in one dimension
 Quantum Information Processing
"... Abstract. This paper treats the quantum random walk on the line determined by a 2 × 2 unitary matrix U. A combinatorial expression for the mth moment of the quantum random walk is presented by using 4 matrices, P,Q,R and S given by U. The dependence of the mth moment on U and initial qubit state ϕ i ..."
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Cited by 59 (28 self)
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Abstract. This paper treats the quantum random walk on the line determined by a 2 × 2 unitary matrix U. A combinatorial expression for the mth moment of the quantum random walk is presented by using 4 matrices, P,Q,R and S given by U. The dependence of the mth moment on U and initial qubit state ϕ is clarified. Furthermore a new type of limit theorems for the Hadamard walk is given. It shows that the behavior of quantum random walk is striking different from that of the classical ramdom walk. 1
Symmetry of distribution for the onedimensional Hadamard walk, Interdisciplinary Information Sciences 10
, 2004
"... Abstract. In this paper we study a onedimensional quantum random walk with the Hadamard transformation which is often called the Hadamard walk. We construct the Hadamard walk using a transition matrix on probability amplitude and give some results on symmetry of probability distributions for the Ha ..."
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Cited by 22 (15 self)
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Abstract. In this paper we study a onedimensional quantum random walk with the Hadamard transformation which is often called the Hadamard walk. We construct the Hadamard walk using a transition matrix on probability amplitude and give some results on symmetry of probability distributions for the Hadamard walk.
Quantum walks: a comprehensive review
, 2012
"... Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting ..."
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Cited by 21 (0 self)
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Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists and engineers. In this paper we review theoretical advances on the foundations of both discrete and continuoustime quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discretetime quantum walks. Furthermore, we have reviewed several algorithms based on both discrete and continuoustime quantum walks as well as a most important result: the computational universality of both continuous and discretetime quantum walks.
Onedimensional discretetime quantum walks on random environments
, 904
"... Abstract. We consider discretetime nearestneighbor quantum walks on random environments in one dimension. Using the method based on a path counting, we present both quenched and annealed weak limit theorems for the quantum walk. 1 ..."
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Cited by 14 (3 self)
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Abstract. We consider discretetime nearestneighbor quantum walks on random environments in one dimension. Using the method based on a path counting, we present both quenched and annealed weak limit theorems for the quantum walk. 1
Quantum random walks in one dimension via generating functions
, 2007
"... We analyze nearest neighbor onedimensional quantum random walks with arbitary unitary coinflip matrices. Using a multivariate generating function analysis we give a simplified proof of a known phenomenon, namely that the walk has linear speed rather than the diffusive behavior observed in classica ..."
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Cited by 14 (3 self)
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We analyze nearest neighbor onedimensional quantum random walks with arbitary unitary coinflip matrices. Using a multivariate generating function analysis we give a simplified proof of a known phenomenon, namely that the walk has linear speed rather than the diffusive behavior observed in classical random walks. We also obtain exact formulae for the leading asymptotic term of the wave function and the location probabilities.
A path integral approach for disordered quantum walks in one dimension
 Fluctuation and Noise Letters
"... Abstract. The present letter gives a rigorous way from quantum to classical random walks by introducing an independent random fluctuation and then taking expectations based on a path integral approach. 1 ..."
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Cited by 13 (6 self)
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Abstract. The present letter gives a rigorous way from quantum to classical random walks by introducing an independent random fluctuation and then taking expectations based on a path integral approach. 1
Limit theorems and absorption problems for quantum random walks
 in one dimension, Quantum Information and Computation
, 2002
"... Abstract. In this paper we consider limit theorems, symmetry of distribution, and absorption problems for two types of onedimensional quantum random walks determined by 2 × 2 unitary matrices using our PQRS method. The one type was introduced by Gudder in 1988, and the other type was studied intens ..."
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Cited by 11 (5 self)
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Abstract. In this paper we consider limit theorems, symmetry of distribution, and absorption problems for two types of onedimensional quantum random walks determined by 2 × 2 unitary matrices using our PQRS method. The one type was introduced by Gudder in 1988, and the other type was studied intensively by Ambainis et al. in 2001. The difference between both types of quantum random walks is also clarified. 1
Absorption problems for quantum walks in one dimension
, 2002
"... Abstract. This paper treats absorption problems for the onedimensional quantum walk determined by a 2 × 2 unitary matrix U on a state space {0,1,...,N} where N is finite or infinite by using a new path integral approach based on an orthonormal basis P,Q,R and S of the vector space of complex 2 × 2 ..."
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Cited by 10 (3 self)
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Abstract. This paper treats absorption problems for the onedimensional quantum walk determined by a 2 × 2 unitary matrix U on a state space {0,1,...,N} where N is finite or infinite by using a new path integral approach based on an orthonormal basis P,Q,R and S of the vector space of complex 2 × 2 matrices. Our method studied here is a natural extension of the approach in the classical random walk. 1
Limit theorems and absorption problems for onedimensional correlated random walks
, 2003
"... In this paper we consider limit theorems and absorption problems for correlated random walks determined by a 2 × 2 transition matrix on the line by using a basis P,Q,R,S of the vector space of real 2 × 2 matrices as in the case of our analysis on quantum walks. ..."
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Cited by 7 (3 self)
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In this paper we consider limit theorems and absorption problems for correlated random walks determined by a 2 × 2 transition matrix on the line by using a basis P,Q,R,S of the vector space of real 2 × 2 matrices as in the case of our analysis on quantum walks.
Absorption problems for quantum random walks in one dimension
, 2002
"... This paper treats absorption problems for the onedimensional quantum random walk determined by a 2 × 2 unitary matrix U on a state space {0,1,...,N} where N is finite or infinite by using a new path integral approach based on an orthonormal basis P,Q,R and S of the vector space of complex 2 × 2 mat ..."
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Cited by 6 (4 self)
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This paper treats absorption problems for the onedimensional quantum random walk determined by a 2 × 2 unitary matrix U on a state space {0,1,...,N} where N is finite or infinite by using a new path integral approach based on an orthonormal basis P,Q,R and S of the vector space of complex 2 × 2 matrices. Our method studied here is a natural extension of the approach in the classical random walk.